As the evocative old Z Cars theme echoes round Goodison Park just before 1.30pm this afternoon, Everton's players walking up the tunnel should be doing so with a rare spring in their step. For once they are fancied to win the Mersey derby, 6-4 favourites with Ladbrokes, for only the second time since 2004. Everton have only been favourites for three of the past 24 League meetings, but now they are peering down from on high at the old enemy.

And yet, and yet; in the accompanying roar from the stands just a hint of apprehension may still be detected. Time after time since Liverpool reunited the teams by winning promotion half a century ago, they have acted as a red nemesis, winning twice as many meetings in that period, including two FA Cup finals, three semi-finals and one League Cup final.

Last season was one of the worst in that respect: defeats by 2-0 and 3-0 in the league, then a Wembley Cup semi-final in which Everton led at half-time through Nikica Jelavic only to lose the game to Luis Suarez and Andy Carroll.

Suarez, according to Everton's captain, Phil Neville, was the player who made the difference and too often for his taste it has been someone in a red shirt who has done so on these occasions. "There's that something extra that you need in these types of games and I think the derbies I've played in, the likes of, say, [Steven] Gerrard has had that little bit extra," he said ahead of today's encounter.

"Of all the derbies I've played in, he's probably been the best player consistently. They've had [Javier] Mascherano over the years, [Fernando] Torres a couple of seasons ago, always that one player who has won it for them. In the semi-final I found it was Suarez."

Honest to a fault, Neville admits that being the form team and favourites today is not necessarily beneficial either: "That game told me it doesn't matter how you play leading up to a derby, it's a one-off game and [about] how you perform. It's a cliché but the form book goes out of the window.

"I think this game is all about belief and having the balls to play. In the games I've played in the Merseyside derby, Liverpool have been better than us apart from two or three games and we have disappointed. That's something we need to put right and that's something the boss is stressing to us."

If Neville takes heart, then it is from looking round the dressing-room and seeing a seasoned squad whose maturity for once compares well with many of their opposite numbers across the park. "We probably shade it in terms of experience now," he said. "They have a new manager, they're building a new journey whereas ours has probably been more consistent and I think this is a time when this team is probably at its peak in terms of having to produce. We now have to produce something or win something and end the season on a note where we've got some success. No excuses. We have a group of players who are at an age where it's now or never in a way.

"For myself, in the last year of my contract, Sylvain [Distin], Jags [Phil Jagielka] playing at his peak, [Leighton] Baines playing at his peak, [Steven] Pienaar... all at the age of 28-32, those peak years. For this group of players it's now or never in terms of success."

Words like "success" and "trophies" still trip off the Neville tongue even after seven years at the club without any. After six Premier League titles, three FA Cups and a Champions' League win with Manchester United, that is something that comes hard, and cannot be assuaged by finishing one place above Liverpool in the table as Everton did last May for only the second time in the Premier League era.

"At Man United, at the end of every season you had a trophy to show for your hard work in a season," he said. "The years you don't win a trophy, you've got an empty feeling. Last season, we were unbelievable from January onwards and everybody finished the season on a high. But I had an empty feeling in my stomach and it's that empty feeling that motivates me. The lap of honour is for people who've won trophies, won medals.

"It has been too long since I won one. We're doing a lap of honour, and rightly so to celebrate with the fans and thank them, but for nothing. And that's what I hate, it lives with me all summer like a churning in my belly."

Liverpool's manager, Brendan Rodgers, was given an early taster of the mania of derby week on the train back from London last Sunday after watching Everton at Queens Park Rangers. "I ended up being in a carriage full of all Everton supporters," he said. "It was an experience. But listen, I know what it means. It is really a unique city. I've only been here a short period of time but the passions run very high. Two wonderful clubs, and I've got to say in the week leading to the Hillsborough report and the vigil, the brilliant support of the Everton supporters has really helped the families."

With three successive clean sheets, Rodgers believes Liverpool are moving in the right direction. What he will demand amid the mayhem of Goodison is cool heads – there have been more red cards than in any other Premier League fixture – and what he calls "that courage to get on the ball, the biggest bravery of all".

Super Money Back Sunday: If Manchester United beat Chelsea today, Ladbrokes will refund all losing first goalscorer, scorecast and correct score stakes as a FREE bet! Phil Neville is a best price of 22-1 to score at any time against Liverpool. Terms and conditions at ladbrokes.com/football